Complete geometric parametrization of the Fisher zeros of the planar Ising model (the complexification conjecture for the Livine–Bonzom–Lis immersion formula)

The partition function of the zero-field Ising model on a finite planar graph $G^*$, written through the high-temperature expansion as the loop polynomial — the generating function of even subgraphs in the edge weights $x_e=\tanh J_e$ — vanishes on a complex hypersurface in $\mathbb{C}^{E^*}$: the Fisher zeros, which in the thermodynamic limit condense onto the critical manifolds. Bonzom and Livine, from the semi-classical asymptotics of 3D quantum gravity, discovered a formula that manufactures such zeros from the geometry of an immersion of the dual triangulation $G$ in three-dimensional Euclidean space: each edge weight is a phase of half the dihedral angle between adjacent face normals times the geometric mean of the tangents of half the two opposite face angles. Lis proved this formula rigorously by constructing a null eigenvector of the Kac–Ward matrix, whose existence is equivalent to the flatness of a natural $\mathrm{SU}(2)$ connection on $G^*$. The catch is that this geometric family has real dimension $|E|-1$, exactly half of the real dimension $2|E|-2$ of the full zero hypersurface. The open problem is to complete the geometric picture: parametrize all Fisher zeros of planar Ising models — conjecturally by a suitable complexification of the immersion geometry, or by a three-dimensional generalization of Chelkak's s-embedding critical weights, with the immersion family as the real slice — and prove completeness. The same sources pose the analogous question for the dimer partition function.

Background

The Ising model on a finite graph $G^*=(U,E^*)$ places spins $\sigma_u=\pm 1$ on the vertices with Hamiltonian $\mathcal H(\sigma)=-\sum_{\{u,u'\}}J_{uu'}\sigma_u\sigma_{u'}$ and partition function $\mathcal Z=\sum_\sigma e^{-\mathcal H(\sigma)}$. Studying $\mathcal Z$ as a function of complexified couplings locates phase transitions: for sequences of graphs of growing size, the zeros of $\mathcal Z$ in the coupling variables — the Fisher zeros, by contrast with the Lee–Yang zeros, which are the zeros in the external-field variable — accumulate on the critical manifolds in the thermodynamic limit. On a finite graph the Fisher zeros form an algebraic hypersurface in the $|E^*|$ complex weight variables, but beyond exactly solved cases little is known about its global structure.

For planar graphs the model is exactly solved. The Kac–Ward formula expresses the squared partition function as the determinant of a finite matrix indexed by directed edges, whose entries carry the turning angles of the planar embedding. Equivalently, the high-temperature expansion rewrites the partition function as the generating function of even subgraphs — subgraphs in which every vertex has even degree:

$$Z_{G^*}(x)\;=\;\sum_{\omega\in\Omega_{G^*}}\prod_{e\in\omega}x_e,\qquad x_e=\tanh J_e,$$

up to an explicit nonvanishing factor. When $G^*$ is trivalent (every vertex has degree three), the even subgraphs are disjoint unions of cycles and $Z_{G^*}(x)$ is called the loop polynomial; the Fisher zeros are the zeros of this polynomial.

A separate line of work ties criticality to discrete geometry. Lis showed that critical Ising weights are encoded by circle patterns — combinatorial embeddings in which each face carries a circle and neighboring circles cross at the vertices. Chelkak's s-embeddings generalize this: from any global solution of the propagation equation for Kadanoff–Ceva fermions (fermionic observables of the Ising model, functions on edges satisfying a local linear propagation relation), one builds an embedding of the graph into Minkowski space $\mathbb{R}^{2,1}$, with an extra origami coordinate tracking a folding of the faces, and the critical weights are expressed through this geometry. These constructions describe critical (infinite-volume) weights but say nothing about the location of the zeros of $Z_{G^*}$ at finite size.

The surprise came from quantum gravity. The Ponzano–Regge state sum — a discrete path integral for three-dimensional Euclidean quantum gravity built from the $6j$-symbols, the recoupling coefficients of irreducible $\mathrm{SU}(2)$ representations — has boundary amplitudes dual to the two-dimensional Ising partition function, and the large-spin asymptotics of the $6j$-symbol, governed by the geometry of a Euclidean tetrahedron, led Bonzom and Livine to a formula for zeros of the loop polynomial on a planar trivalent graph $G^*$ in terms of an oriented immersion of the dual triangulation $G$ in $\mathbb{R}^3$ (each face mapped to a nondegenerate triangle): if $u,u'$ are adjacent faces sharing the primal edge $\{v_1,v_2\}$, then

$$\tanh \mathcal J_{\{u,u'\}}\;=\;\exp\!\Big(\frac{i\,\vartheta_{uu'}}{2}\Big)\sqrt{\tan\frac{\varphi_{uu'}}{2}\,\tan\frac{\varphi_{u'u}}{2}},$$

where $\varphi_{uu'}\in(0,\pi)$ is the face angle of the triangle $u$ at the vertex opposite the shared edge (and $\varphi_{u'u}$ likewise for $u'$), and $\vartheta_{uu'}\in(-\pi,\pi]$ is the oriented angle between the face normals of $u$ and $u'$. Theorem (Livine–Bonzom, conjectured; Lis, proved): with these weights, $Z_{G^*}=0$. Different immersions give different zeros; when all triangles are coplanar the weights are positive and reduce to the critical circle-pattern weights. Garay and Livine verified the formula numerically on random polyhedra; Lis proved it rigorously. The proof identifies the Kac–Ward transition data, after a change of coordinates, with an $\mathrm{SU}(2)$ connection on the faces of $G$: a null eigenvector of the Kac–Ward matrix exists exactly when this connection is flat (holonomy — the group element accumulated by transporting around a loop — trivial around every loop), and flatness is equivalent to the existence of the immersion, since $\mathrm{SU}(2)$ double-covers the rotations of $\mathbb{R}^3$. The proof also extends to graphs on the torus and higher-genus surfaces, where the Kac–Ward determinant computes an even-subgraph generating function with extra signs depending on the homology class, and the same weights make that signed partition function vanish.

The gap is a dimension count. An immersion of the triangulation $G$ is fixed, up to rigid motions, by the $n=|E|$ edge lengths — $n-1$ real parameters — so the geometric family of zeros has real dimension $n-1$. The zero set of $Z_{G^*}$, defined by one complex equation in $n$ complex variables, is a hypersurface of complex dimension $n-1$, i.e. real dimension $2n-2$. The formula thus captures only half of the Fisher zeros (for the tetrahedral graph: a 5-parameter family inside a 10-real-dimensional zero set). Bonzom and Livine conjecture that the whole zero set is obtained by a suitable complex continuation of their formula — a geometric representation that would in particular illuminate the Kramers–Wannier duality (the low-temperature/high-temperature self-duality of the two-dimensional Ising model mapping strong couplings to weak couplings on the dual graph), under which the known real formula is not stable. Lis expects the missing class of zeros to come from a three-dimensional generalization of the s-embedding critical weights, which in the flat case generalize the circle-pattern critical weights. Finally, since the planar Ising partition function is itself a dimer partition function on a decorated graph (the dimer model counts perfect matchings and is exactly solvable by Pfaffians of Kasteleyn matrices — weighted adjacency matrices with orientation signs chosen so that the Pfaffian equals the dimer partition function — on planar graphs), the same sources pose the analogous question of a three-dimensional geometric description of the zeros of the dimer partition function.

Problem Statement

In the proven base setting, let $G$ be a finite connected planar triangulation with $n=|E|$ edges and $G^*$ its trivalent planar dual, and let

$$Z_{G^*}(x)\;=\;\sum_{\omega\in\Omega_{G^*}}\prod_{e\in\omega}x_e$$

be the loop polynomial (the Ising partition function in the weights $x_e=\tanh J_e$, up to a nonvanishing factor). Its zero set $V=\{x\in\mathbb{C}^{E^*}:Z_{G^*}(x)=0\}$ — the Fisher zeros — is a hypersurface of complex dimension $n-1$. The theorem of Livine–Bonzom and Lis produces, from every oriented immersion of $G$ in $\mathbb{R}^3$, a zero $y=\tanh\mathcal J(\mathbf x)$ with $\tanh\mathcal J_{\{u,u'\}}=e^{i\vartheta_{uu'}/2}\sqrt{\tan(\varphi_{uu'}/2)\tan(\varphi_{u'u}/2)}$, where $\varphi_{uu'}\in(0,\pi)$ is the face angle of the immersed triangle $u$ at the vertex opposite the shared primal edge ($\varphi_{u'u}$ likewise for $u'$) and $\vartheta_{uu'}\in(-\pi,\pi]$ is the oriented angle between the face normals of $u$ and $u'$; this family has real dimension $n-1$ — half of the real dimension of $V$.

The problem, as posed by the sources, is to complete this geometric picture:

(i) Construct an explicit parametrization of the full Fisher zero set $V$ for planar Ising models by geometric data generalizing the immersion construction, and prove it complete over the class of finite connected planar graphs: for every finite connected planar graph $G^*$, every zero of $Z_{G^*}$ arises (the trivalent duals of triangulations being the base setting, extended to duals of higher vertex degree, in which the immersion maps the vertices around each face of $G$ to a single circle in $\mathbb{R}^3$, as in Remark 1 of Lis's paper).

(ii) Realize the parametrization of (i) through one of the two conjectured mechanisms: either a suitable complexification of the three-dimensional immersion geometry (Bonzom–Livine), or a three-dimensional generalization of the s-embedding critical weights (Lis), in either case with the proven immersion family appearing as the real geometric slice, and with the dependence on the geometric data local per edge in the same sense as the known formula. For a positive resolution, both (i) and (ii) are required: a complete parametrization obtained by a mechanism other than the two named mechanisms does not resolve the task.

A stated motivation for the complexification route (Bonzom–Livine) is that a consistent complexification would in addition provide a representation of the Kramers–Wannier low-temperature/high-temperature duality on the zero set, under which the known real formula is not stable; this duality representation is a downstream consequence expected of a complete parametrization, not an independent deliverable.

A rigorous refutation also answers the question: a proof that the loop-polynomial hypersurfaces admit no parametrization of the required geometric form — an impossibility result covering all geometric families of the kind described in (ii), not merely one chosen family (for instance, an explicit characterization of a class of Fisher zeros provably outside every such family). The same sources additionally pose the companion question of a three-dimensional geometric description of the zeros of the dimer partition function; that companion question is recorded here as context and as the natural extension, but the completion boundary of this problem is the Ising objective (i)–(ii).

References

1. Marcin Lis, "Zeros of Planar Ising Models via Flat SU(2) Connections", Communications in Mathematical Physics 407(1), Article 6 (2026). DOI 10.1007/s00220-025-05516-x; arXiv:2409.19639. https://arxiv.org/abs/2409.19639
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